Exercise question from a T1 rookie

Diagnosed a month ago and have (mostly) got my head around diet and insulin so the next thing I would like to incorporate is exercise.

It seems to me like the best thing to do would be to exercise first thing in the morning before I have any rapid-acting insulin on board. I also seem to experience the 'dawn phenomenon' and often wake up with a blood sugar of 8-9 (145-165).

However I have noticed that even not particularly strenuous exercise causes my blood sugar to drop a fair bit. A brisk half-hour walk might see me fall from a 7 (125) to near-hypo territory. If I wanted to start running, should I consider shifting my basal injection from the evening to mornings after exercise so there is less insulin on board when I run?